An elementary counterexample to the finiteness conjecture

Blondel, Vincent;Theys, Jacques;Vladimirov, Alexander A.
(2003) SIAM Journal on Matrix Analysis and Applications — Vol. 24, n° 4, p. 963-970 (2003)

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  • Blondel, Vincentorcid-logoUCLouvain
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  • Theys, JacquesUCLouvain
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  • Vladimirov, Alexander A.Russian Academy of Science
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Abstract
We prove that there exist ( infinitely many) values of the real parameters a and b for which the matrices [GRAPHICS] have the following property: all infinite periodic products of the two matrices converge to zero, but there exists a nonperiodic product that doesn't. Our proof is self-contained and fairly elementary; it uses only elementary facts from the theory of formal languages and from linear algebra. It is not constructive in that we do not exhibit any explicit values of a and b with the stated property; the problem of finding explicit matrices with this property remains open.
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Blondel, V., Theys, J., & Vladimirov, A. A. (2003). An elementary counterexample to the finiteness conjecture. SIAM Journal on Matrix Analysis and Applications, 24(4), 963-970. https://doi.org/10.1137/S0895479801397846 (Original work published 2003)